Class 12 Continuity And Differentiability · Long Answer3 Marksbranch_sensitive_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #139
Differentiate y=arccos[2^(x+1)/(1+4^x)] on its differentiable real intervals.
✅ Step-by-Step Solution
Final Answer: −2^(x+1)ln2/(1+4^x) for x<0; +2^(x+1)ln2/(1+4^x) for x>0; not differentiable at 0.
- Put u=2^x>0. Then the argument is 2u/(1+u²)=sin(2arctan u).
- Since 0<2arctan u<π, the principal arccosine equals |π/2−2arctan u|. It changes branch at u=1, or x=0.
- Differentiate π/2−2arctan u for x<0 and 2arctan u−π/2 for x>0. The magnitude is 2u ln2/(1+u²). Opposite slopes at 0 produce a corner.
- Thus y′ has the stated negative and positive branches, and y′(0) does not exist.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put u=2^x>0. Then the argument is 2u/(1+u²)=sin(2arctan u).
- [1 mark] Since 0<2arctan u<π, the principal arccosine equals |π/2−2arctan u|. It changes branch at u=1, or x=0.
- [1 mark] Differentiate π/2−2arctan u for x<0 and 2arctan u−π/2 for x>0. The magnitude is 2u ln2/(1+u²). Opposite slopes at 0 produce a corner.
- [0.5 marks] Thus y′ has the stated negative and positive branches, and y′(0) does not exist.
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